Transcript Evoluciones numericas de agujeros negros binarios

Solucion numerica de las
ecuaciones de Einstein:
Choques de agujeros negros
Jose Antonio Gonzalez
IFM-UMSNH
25-Abril-2008
ENOAN 2008
Overview
• Introduction
– Binary black hole problem
• Some ingredients
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3+1 decomposition
Formulation of the equations
Initial data
Gauge
Mesh refinement
Boundary conditions
Excision
Diagnostic tools
• Applications
• Conclusions
The big picture
Physical System
describes
Model
GR (numerical relativity)
PN
Perturbation theory
Non-GR?
Detectors
observe
implications
External Physics
Astrophysics
Fundamental physics
Cosmology
Numerical relativity
-Two 10 solar mass black holes
-Frequency ~ 100Hz
-Distort the 4km mirror spacing
by about 10^-18 m

3+1 decomposition
GR: “Space and time exist together as Spacetime’’
Numerical relativity: reverse this process!
ADM 3+1 decomposition
3-metric
lapse
shift
 ij
Arnowitt, Deser, Misner (1962)
York (1979)
Choquet-Bruhat, York (1980)

i
lapse, shift Gauge
Einstein equations
 6 Evolution equations
4 Constraints
Constraints preserved under evolution!
ADM equations
Evolution equations
(t  L ) ij  2Kij
(t  L )Kij  Di Dj  [Rij  2KimK m j  Kij K ]
Constraints
R  K 2  Kij K ij  0
 Dj K ij  Di K  0
Evolution
Solve constraints initially
Evolve data
Reconstruct spacetime
Extract physics
Formulation of the equations
ADM: unsuccessful;
weakly hyperbolic!
Balance laws:
Bona, Massó (H-code)
BSSN (most popular)
Shibata, Nakamura ‘95
Baumgarte, Shapiro ‘99
Split degrees of freedom (similar to initial data split)
Hyperbolicity
Sarbach et.al.‘02;
Gundlach, Martin-Garcia
Generalized harmonic formulation Garfinkle ‘04
Harmonic gauge well-posed
Choquet-Bruhat ‘62
Wave equations for g
BBH-breakthrough
Pretorius ‘05
KST, NOR,…
Many more: ADM-like family:
Z4
Harmonic family:
Control of constraints: LSU, Caltech, Gundlach
The BSSN formulation
Initial data
Two difficulties: Constraints, realistic data
 ij   4~ij
York-Lichnerowicz split
Conformal transverse traceless
Physical transverse traceless
Thin sandwich
1
K ij  Aij   ij K
3
York, Lichnerowicz
O’Murchadha, York
Wilson, Mathews; York
Conformal flatness:
Spurious radiation does not seem problematic, but alternatives studied
Generalized analytic solutions: Isotropic Schwarzschild
 Time symmetric, N -holes: Brill-Lindquist, Misner (1960s)
 Spin, linear momenta:
Bowen, York (1980)
 Punctures
Brandt, Brügmann (1997)
Excision data: Isolated Horizon condition on excision surface
Meudon group; Cook, Pfeiffer; Ansorg
Quasi-circularity:
Effective potential method
PN fit
helical killing vector
Gauge
Specific problem in GR: Coordinates constructed during evolutions
Einstein equations say nothing about  ,  i
i
Highly non-trivial: Prescribe  ,  to avoid coordinate singularities
Maximal slicing,
min.distortion shift
Smarr, York ‘78
Driver conditions
Balakrishna
et.al.’96
~
1+log, -driver
AEI
Drive to stationarity
Moving punctures
UTB, Goddard ‘06
special
case
Harmonic coords
Choquet-Bruhat‘62
Analytic studies
Bona-Massó family
Bona, Massó ‘95
special
case
Study singularity
avoidance
Alcubierre ‘03
Generalized harmonic
Garfinkle ‘04
Pretorius ‘05
gauge sources
i
relation to  , 
Mesh-refinement, boundaries
 1M
3 length scales: BH
Wave length  10M
Wave zone  100M
Choptuik ’93
AMR, Critical phenomena
Stretch coordinates: Fish-eye
Lazarus, AEI, UTB
FMR, Moving boxes: Berger-Oliger
BAM
Brügmann’96
Carpet Schnetter et.al.’03
AMR: Steer resolution via scalar
Paramesh:
MacNeice et.al.’00, Goddard
modified Berger-Oliger:
Pretorius, Choptuik ’05
SAMRAI
Refinement boundaries: reflections, stability
Lehner, Liebling, Reula ‘05
Outer boundary conditions
Problems: Well-posedness of equations?
Constraint violations?
BCs that satisfy constraints and/or well-posedness
Friedrich, Nagy ‘99
Calabrese, Lehner, Tiglio ‘02
Frittelli, Goméz ‘04
Sarbach, Tiglio ‘04
Kidder et.al.‘05, Lindblom et.al.‘06
Tested with success in BBH simulation: Lindblom et.al.‘06
Conformal, null-formulation: Untested in BBH simulations
Compactification in 3+1 Pretorius ‘05
Push boundaries “far out”, use Sommerfeld condition
Used successfully by most groups; accuracy limits?
Multi-patch approach: Efficiency
AEI (Cactus): Thornburg et.al.: excision, Char.Code
LSU (below), Austin (below), Cornell-Caltech (below)
Black hole excision
Cosmic censorship: Causal disconnection of region inside AH
Unruh ’84 cited in Thornburg ‘87
Grand Challenge: Causal differencing
“Simple Excision”
Alcubierre, Brügmann ‘01
Dynamic “moving” excision
Pitt-PSU-Texas
PSU-Maya
Pretorius
combined with Dual coordinate frame Caltech-Cornell
Mathematical properties: Wealth of literature
Diagnostic tools
A computer just gives numbers! These are gauge dependent!
Convert to physical information…
ADM mass, momentum
Arnowitt, Deser, Misner ‘62
Bondi mass, News function (Characteristic approach)
Gravitational Waves
Zerilli-Moncrief formalism
Newman-Penrose scalar 4   tt h  itt h
Black hole quantities: mass, momentum, spin, area,…
Apparent Horizon
Alcubierre, Gundlach (Cactus)
Schnetter ‘03
Thornburg ‘03 (AHFinderDirect)
Pretorius
Event horizon
Diener ‘03
Isolated, Dynamic Hor.
Ashtekar, Krishnan ’03
Ashtekar et.al.
Dreyer et.al. ’02
2004
How far we are?
2007
Spinning holes: The orbital hang-up

 Spins alligned with L  inspiral delayed, Erad , J rad
larger
No extreme
Kerr holes
produced

  Spins anti-alligned with L  inspiral fast Erad , J rad smaller

Gravitational recoil
Anisotropic emission of GW carries away linear momentum
 recoil of remaining system
Merger of galaxies
 Inspiral and merger of black holes
 Recoil of merged hole
 Displacement, Ejection?
Astrophysical relevance
BH inspiral  kick  possible ejection of BH from host
30 km/s
Escape velocities: globular clusters
20  100 km/s
dSph
100  300 km/s
dE
large galaxies
 1000 km/s
Merritt et al.’04
Non-spinning binaries
Emerging picture: Kicks unlikely to exceed a few 100 km/s
Numerical relativity allows accurate estimates
Campanelli ’05
Herrmann et al.’06
Baker et al.’06
Close limit calculations
Sopuerta et al.’06 a,b
Upper and lower bounds
Including eccentricity increases kick
vkick  (1  e)
for small eccentricities
EOB approximation: account for deviations from Kepler law
Damour & Gopakumar ‘06
Non-spinning binaries
Higher order PN
Blanchet et al.’05
Systematic parameter study Gonzalez et al.’06
Moving puncture method
BAM code
Nested boxes, resolutions h  ms /40
Extract  4 calculate linear momentum
Vary mass ratio: q  1 : 1...1 : 4,
  0.25...0.16
150,000 CPU hours
Non-spinning binaries: Maximal kick
Maximal kick: 175 .7  11 km/s at   0.195 0.005
Recoil of spinning binaries
Kidder ’95: PN study including recoil of spinning holes
= “unequal mass” + “spin(-orbit)”
Penn State ‘07: Spin-orbit term larger
a
 0.2,...,0.8
m
extrapolated: v  475 km/s
AEI ’07:
extrapolated: v  440 km/s
a1
 0 .6
m
a2
 0.0,...,0.6
m
Recoil of spinning binaries
UTB-Rochester
v  454 km/s
maximum predicted: v  1300 km/s
NASA Goddard:
vtrans  15  30 km/s Spin effect
vlong  2  5 km/s
Unequal-mass effect
PN predictions remarkably robust
Fitting formulas
Getting even larger kicks
Trajectories:
Discretization error: v  43 km/s
Dependence on Extraction radius
Error fall-off: v  120 km/s
Reducing eccentricity
Data analysis and PN comparisons
Since it is expensive to generate an entire physical bank of
templates using numerical simulations, it is better to construct
a phenomenological bank –unequal mass, non spinning black
holes-
Thick red line  NR waveforms
Dashed black  ‘best matched’ 3.5 PN waveforms
Thin green  Hybrid waveforms
Eccentricity
IMRI’s:
Motivation
• Stellar mass black holes  M~1-10 Msun
• Intermediate mass bh’s  M~102-4 Msun
• Supermassive bh’s  M~106-9 Msun
Why IMRIs and EMRIs are interesting?
•Astrophysics
•Data Analysis and gravitational waves detection:
Gravitational waves emited during the merger of stellar-mass
black holes into a IMBHs will lie in the frequencies of Advanced
LIGO (Brown et al. 2007)
•Tests of General Relativity
•Comparison with PN and perturbation theory
•Numerical simulations are expensive
•How many orbits are required?
Data analysis 10? 100?
 Compare with PN!
•How far we need to go in mass ratios?
1:100? 1:1000???
 Hopefully not!
Mass ratio 1:10
Parameters:
• M1 = 0.25 , M2 = 2.5 , M = M1+M2
• D = 19.25 = 7M
• q = M1/M2 = 10 , η = q/(1+q)2 = 0.0826
•Problems:
–Gauge:
–Resolution:
[η]= 1/M
Kick
V~62 km/s
Fitchett (MNRAS 203 1049,1983)
Gonzalez et al. (PRL 98 091101, 2007)
Radiated energy
ΔE/M=0.580192 η2
ΔE/M~0.004018
Berti et al. (2007)
Final spin
aF/MF~0.2602
Damour and Nagar (2007)
Energy distribution
ERAD = 0.011001
l=2 75.62%
l=3 16.36%
l=4
4.96%
l=5
1.74%
Conclusions
• After a lot of work and effort….it seems to
work!
• It is over? No way!
– It is necessary to improve accuracy
– Now it is possible to do physics –the original
purpose of everything– Data analysis
– Parameter estimations
• Matter